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On non-commuting graph of a finite ring

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arxiv 1703.05039 v1 pith:XMCKTZ6L submitted 2017-03-15 math.RA

On non-commuting graph of a finite ring

classification math.RA
keywords finitegammagraphnon-commutingringgraphsisomorphicrings
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The non-commuting graph $\Gamma_R$ of a finite ring $R$ with center $Z(R)$ is a simple undirected graph whose vertex set is $R \setminus Z(R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab \ne ba$. In this paper, we show that $\Gamma_R$ is not isomorphic to certain graphs of any finite non-commutative ring $R$. Some connections between $\Gamma_R$ and commuting probability of $R$ are also obtained. Further, it is shown that the non-commuting graphs of two $\mathbb{Z}$-isoclinic rings are isomorphic if the centers of the rings have same order

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